Frustrated by the light clock special relativity thought experiment

Here is this age old thought experiment being told by a professor on Sixty Symbols: https://youtu.be/Cxqjyl74iu4
This explanation using the light clock is extremely frustrating. How can one use a hypothetical example which is physically impossible and then say the "result" explains SR? The photon would never hit the top mirror directly above its source b/c light does not take on the velocity of its source. Instead, the instant it leaves its source it goes straight up while the rocket moves forward, and would strike the back of the rocket (or the top somewhere to the left of the mirror). If the photon struck the mirror it would not move forward with the rocket, but again would go straight down while the rocket moves forward, b/c for the photon to move forward it woud have to feel the friction of the mirror pushing it forward, which is again impossible. The reason a wave such as sound would have the trajectory shown in this example is that the medium inside the rocket, air, is moving at the speed of the rocket and the sound wave would take on that velocity as it left its source. Light does not use a medium to move. The reason a physical object such as a ball would have the trajectory shown is that particles take on the velocity of the souce that is accelerating them. Again, light does not take that velocity on, but instead it instantly has its standard speed (c) as it leaves its source. So, the photon does the exact same thing leaving a moving source as it would a stationary source, it moves at the speed of light in the direction it's facing, hence no length is added to its trajectory as stated in the example, and thus does not prove the time dilation of SR. Anybody else hearing me here? Thoughts?